Question

represent -2048.75 in IEEE-754 single precision format.

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Answer 1

Converting 2048.75 to binary
   Convert decimal part first, then the fractional part
   > First convert 2048 to binary
   Divide 2048 successively by 2 until the quotient is 0
      > 2048/2 = 1024, remainder is 0
      > 1024/2 = 512, remainder is 0
      > 512/2 = 256, remainder is 0
      > 256/2 = 128, remainder is 0
      > 128/2 = 64, remainder is 0
      > 64/2 = 32, remainder is 0
      > 32/2 = 16, remainder is 0
      > 16/2 = 8, remainder is 0
      > 8/2 = 4, remainder is 0
      > 4/2 = 2, remainder is 0
      > 2/2 = 1, remainder is 0
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 100000000000
   So, 2048 of decimal is 100000000000 in binary
   > Now, Convert 0.75000000 to binary
      > Multiply 0.75000000 with 2.  Since 1.50000000 is >= 1. then add 1 to result
      > Multiply 0.50000000 with 2.  Since 1.00000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.75 of decimal is .11 in binary
   so, 2048.75 in binary is 100000000000.11
-2048.75 in simple binary => 100000000000.11
so, -2048.75 in normal binary is 100000000000.11 => 1.0000000000011 * 2^11

single precision:
--------------------
sign bit is 1(-ve)
exp bits are (127+11=138) => 10001010
   Divide 138 successively by 2 until the quotient is 0
      > 138/2 = 69, remainder is 0
      > 69/2 = 34, remainder is 1
      > 34/2 = 17, remainder is 0
      > 17/2 = 8, remainder is 1
      > 8/2 = 4, remainder is 0
      > 4/2 = 2, remainder is 0
      > 2/2 = 1, remainder is 0
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10001010
   So, 138 of decimal is 10001010 in binary
frac bits are 00000000000110000000000

so, -2048.75 in single-precision format is 1 10001010 00000000000110000000000
in hexadecimal it is 0xC5000C00